Section 2-4: Reasoning in Algebra
TPI 32A: apply reflective, transitive, or symmetric prooperties of
equality or congruence
Objectives:
• Connect reasoning in algebra and geometry
• Justify steps in deductive reasoning
In geometry
• postulates, definitions, & properties are accepted as true
• you use deductive reasoning to prove other statements
We will look at some basic properties used to justify
statements…..
….. which leads to writing proofs.
Addition Property of Equality
If a = b, then a + c = b + c
Add same amount to both sides of an equation.
Subtraction Property of Equality
If a = b, then a - c = b - c
Subtract same amount to both sides of an equation.
Multiplication Property of Equality
If a = b, then a ∙ c = b ∙ c
Multiply both sides of an equation by the same amount.
Division Property of Equality
If a = b and c  0, then a b

c c
Divide both sides of an equation by the same amount.
Reflective Property of Equality
a=a
Ex: 5 = 5
Symmetric Property of Equality
Ex: 3 = 2 + 1 and 2 + 1 = 3 are the same.
If a = b, then b = a
Transitive Property of Equality
If a = b and b = c, then a = c.
EX: If 3 + 4 = 7 and 5 + 2 = 7, then 3 + 4 = 5 + 2.
Substitution Property of Equality
If a = b , then b can replace a in any expression.
Ex: a = 3; If a = b, then 3 = 3.
Distributive Property
a(b + c) = ab + ac
Ex: 3(x + 3) = 3x + 9
Using Properties to Justify Steps in Solving Equations
Algebra Solve for x and justify each step.
Given: m AOC = 139
m AOC = 139
Given
m AOB + m BOC = m AOC Angle Addition Postulate
x
+
2x + 10 = 139
Substitution Property
3x + 10 = 139
3x = 129
Simplify
x = 43
Division Property of Equality
Subtraction Property of Equality
Using Properties to Justify Steps in Solving Equations
Solve for x and justify each step.
Given: LM bisects KLN
LM bisects KLN
MLN = KLM
4x = 2x + 40
2x = 40
x = 20
Given
Def of Angle Bisector
Substitution Property
Subtraction Property of Equality
Division Property of Equality
Using Properties to Justify Steps in Solving Equations
Solve for y and justify each step
Given: AC = 21
AC = 21
Given
AB + BC = AC
2y + 3y - 9 = 21
Segment Addition Postulate
Substitution Property
5y – 9 = 21
5y = 30
Simplify
y=6
Division Property of Equality
Addition Property of Equality
Find AB and BC by substituting y = 6 into the expressions.
The Reflective, Symmetric, and Transitive Properties of
Equality have corresponding properties of congruence
that can be used to justify statements.
Reflective Property of Congruence
AB  AB
A  A
Symmetric Property of Congruence
If AB  CD, then CD  AB.
If A  B, then B  A
Transitive Property of Congruence
If AB  CD and AB  EF, then CD  EF.
If A  B and B  C, then A  C.
Using Properties of Equality and Congruence
Name the property of congruence or equality the justifies
each statement.
a. K  K
Reflective Property of 
b. If 2x – 8 = 10, then 2x = 18 Addition Property of Equality
c. If RS  TW and TW  PQ,
then RS  PQ.
Transitive Property of 
d. If m A = mB, then
m B = mA
Symmetric Property of Equality
Use what you know about transitive properties
to complete the following:
The Transitive Property of Falling Dominoes:
If domino A causes domino B to fall, and domino B
causes domino C to fall, then domino A causes domino
C
_______
to fall.